Extending Predictive Capabilities to Network Models
Pål-Eric Øren, Stig Bakke, and Ole Jakob Arntzen
We reconstruct 3-D sandstone models which give a realistic description of the complex pore space
observed in actual sandstones. The reconstructed pore space is transformed into a pore network
which is used as input to a two-phase network model. The model simulates primary drainage and
water injection based on a physical scenario for wettability changes at the pore level. We derive
general relationships between pore structure, wettability, and capillary pressure for the different
pore level displacement mechanisms which may occur in the network model.
We present predicted transport properties for three different reconstructed sandstones of
increasing complexity: Fontainebleau, a water wet Bentheimer, and a mixed wet reservoir rock.
Predicted transport properties are in good agreement with available experimental data. For the
reservoir rock, both the experiments and the simulated results show that continuos oil films allow
low oil saturations to be reached during forced water injection. However, the oil relative
permeability is very low.
Introduction
The microstructure of a porous medium and the physical
characteristics of the solid and the fluids which occupy the
pore space determine several macroscopic properties of the
medium. These properties include transport properties of
interest such as permeability, formation factor, relative
permeability, and capillary pressure. In principle, it should
be possible to determine these properties by appropriately
averaging the equations describing the physical processes
occurring on the pore-scale. The prediction of macroscopic
transport properties from the associated pore-scale
parameters is a long-standing issue which has been the
subject of much investigation. One commonly applied tool in
this investigation is the network model.
The premise of the network model is that the void space
of a porous medium can be represented by a network of
interconnected pores in which larger pores (pore bodies) are
connected by smaller pores (pore throats). Since the
pioneering work of Fatt1-3, network models have been used
extensively to study different displacement processes in
simple or idealised porous media4-20. Seldom, however, do
such models claim to be representative of reservoir rocks21.
The extension of network modeling techniques to real
porous media is hampered by the difficulty of adequately
Copyright 1998, Society of Petroleum Engineers, Inc.
Paper first presented at the SPE Annual Technical Conference & Exhibition held in San
Antonio, U.S.A., 5-8 October, 1997.
Received for review, November 23, 1995
Revised, April 1, 1996
Accepted for publication, June 15, 1996
Camera-ready copy received, October 14, 1996
SPE 38880
describing the complex nature of the pore space. Advanced
techniques such as microtomographic imaging21-25 and serial
sectioning26-27 provide a detailed description of the pore
space at micrometer resolution. In practice, however,
information about the microstructure of reservoir rocks is
limited to 2-D thin section images and to pore throat entry
sizes determined from mercury injection. These data are
insufficient to directly construct a 3-D pore network which
replicates the microstructure of the medium. As a result,
simplifying assumptions about the topology and geometry of
the pore structure must be invoked.
Despite these simplifications, network models have
proved to be powerful tools for extrapolating limited
measured data and for developing valuable insight into
complex multiphase flow phenomena such as capillary
pressure and relative permeability hysteresis12,28, the effect of
wettability29-32, and three-phase flow33-38. However, the
difficulty in adequately describing the pore network of
reservoir rocks has prevented network models from being
used as a predictive tool, thus greatly limiting their
application in the oil industry.
In the present work, geostatistical information obtained
from image analyses of 2-D thin sections are used to generate
a reliable reconstruction of the complex rock-pore system in
3-D. The network representation of the pore space is
constructed from topological and geometrical analyses of the
fully characterised reconstructed sample. The pore network
is used as input to a two-phase network model which
simulates drainage and water injection based on a physical
model for wettability alteration on the pore level. Predicted
transport properties for different reconstructed sandstones
are compared with experimental data.
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
Experimental
The experimental measurements reported here were
performed on outcrop and on reservoir sandstones. The
outcrop rock consisted of three homogenous core samples
which were cut from a single block of Bentheimer quarried
sandstone. This strongly water wet sandstone is well-sorted
and composed mainly of quartz (70-80%), feldspar (2025%), and authigenic clays (2-3%). The main clay
component is pore-filling kaolinite. The reservoir rock
consisted of two preserved mixed wet core plugs. This North
Sea Lower Brent reservoir rock is a micaceous (10-12%)
feltspatic (15-19%) sandstone with abundant pore-filling
kaolinite. Petrophysical properties for the different plug
samples are given in Table 1.
The Bentheimer plugs were tested at room temperature
using synthetic brine (3% NaCl) and a paraffinic oil.
Measurements on the reservoir core plugs were performed at
70oC using simulated formation brine and a dead crude oil.
Fluid properties at the specific test conditions are
summarised in Table 2.
TABLE 1—PETROPHYSICAL PROPERTIES FOR
BENTHEIMER (B) AND RESERVOIR ROCK (R) SAMPLES
Sample
B1
B2
B3
R1
R2
Porosity
0.232
0.241
0.237
0.297
0.293
ρ (g/cm )
2.640
2.650
2.647
2.67
2.67
3
kw (mD)
2840
2820
2930
358
326
FRF
11.6
12.1
12.0
imbibed both water and oil and the Amott wettability indices
(0.08 and 0.1) indicate that the samples are mixed wet.
TABLE 3—PRIMARY DRAINAGE CAPILLARY PRESSURE
DATA FOR THE BENTHEIMER SAMPLES
Water Saturation
Pc (kPa)
0
2.5
5.0
10.0
20.0
40.0
80.0
150.0
300.0
500.0
Sample B1
1.0
1.0
0.692
0.135
0.090
0.073
0.063
0.056
0.054
0.051
Sample B2
1.0
0.972
0.648
0.137
0.098
0.080
0.068
0.062
0.057
0.052
Sample B3
1.0
0.956
0.423
0.129
0.093
0.079
0.068
0.063
0.058
0.055
TABLE 4—CAPILLARY PRESSURE DATA FOR THE
RESERVOIR ROCK SAMPLES
Forced Water Injection
Pc (kPa)
0.0
-1.8
-4.1
-7.3
-13.8
-32.9
-71.1
-139.5
-278.9
Sw (R1)
0.403
0.688
0.763
0.802
0.820
0.876
0.901
0.918
0.950
Sw (R2)
0.441
0.711
0.771
0.817
0.840
0.890
0.936
0.936
0.950
Secondary Drainage
Pc (kPa)
0.0
0.9
2.1
4.7
11.4
33.5
75.3
145.2
284.8
Sw (R1)
0.774
0.610
0.343
0.232
0.208
0.176
0.172
0.167
0.162
Sw (R2)
0.771
0.608
0.349
0.241
0.202
0.193
0.189
0.188
0.188
TABLE 2—TEST CONDITIONS AND FLUID PROPERTIES
Property
temperature
brine density
brine viscosity
oil density
oil viscosity
oil-water IFT
Bentheimer
o
20 C
3
1.02 g/cm
1.06 cp
3
0.76 g/cm
1.40 cp
35.0 mN/m
Reservoir Rock
o
70 C
3
0.99 g/cm
0.43 cp
3
0.83 g/cm
3.11 cp
7.1 mN/m
Capillary Pressure. Primary drainage capillary pressure
functions for the Bentheimer core plugs were measured by
the porous plate method. A ceramic porous plate with a gaswater threshold pressure of 15 bar was installed on the lower
end-face of the vertically oriented sample. Water production
was measured at nine different drainage pressures in the
range zero to 5 bar. The measured capillary pressure data
were similar for all three samples and are summarised in
Table 3.
Waterflood and secondary drainage capillary pressure
functions for the reservoir core plugs were determined from
centrifuge displacements. Fluid saturations were calculated
from material balance and Karl Fischer analysis at the end of
the experiments. The capillary pressure data for the two
samples were similar and are tabulated in Table 4.
Spontaneous imbibition of water (oil) were measured prior to
the waterflood (secondary drainage) and used to calculate
Amott wettability indices. Both core plugs spontaneously
Relative Permeability. Primary drainage and waterflood
relative permeability curves for the Bentheimer plugs were
determined by the steady state method. The measurements
were performed on vertically oriented samples. The total
flowrate (200 cm3/hr) was constant during the experiments.
Steady state relative permeability measurements were
obtained by keeping the oil and water flowrates constant
until the conditions in the core stabilised. The pressure drop
across the core was monitored continuously using a
differential pressure transducer. Relative permeability to
each phase was calculated using Darcy’s law.
Oil relative permeabilities for the reservoir rock during
forced water injection (i.e., negative capillary pressure) were
determined from a single speed (2500 RPM = -70kPa)
centrifuge displacement applying Hagoort’s analytical
method39 to compute relative permeability. Fluid production
was monitored using an automated centrifuge system
(CENTAS). Production data were recorded every two
seconds at the beginning of the displacement and then at
increasing intervals up to every five minutes. The experiment
run for 14 hours.
Thin Section Analysis. Thin sections were prepared from
the end pieces of the core plugs used in the experiments.
Back-scattered SEM (BSE) images of the thin sections were
binarised and analysed using a KONTRON KS400 image
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
analyser. An erosion-dilation algorithm was used to partition
the sandgrain matrix into discrete grains. We measured the
diameter of each grain and constructed a grain-size
distribution curve. Cathodoluminescence images were used
to distinguish quartz cement from the original detrital grains.
The measured grain diameter is correct only if the thin
section cuts through the centre of the grain. The ratio of the
measured grain diameter to the quartz cement thickness was
used to determine whether or not the section cuts close to the
grain centre. If this ratio is below a pre-defined cut-off value,
which depends on the degree of quartz cementation, the grain
diameter is assumed to be non-representative and the grain is
removed from the grain-size distribution curve.
The porosity of the sample is defined as the fraction pore
area contained in the BSE image. It is determined by first
thresholding the epoxy-filled pore space from the rock matrix
and then measure the corresponding area. The total amount
of clay is determined similarly. The BSE-signals from clays
fall in-between the epoxy and quartz/feldspar signals. By
thresholding the clay one may thus measure the total amount
of clay in the image. The clay texture is determined visually.
Sandstone Reconstruction
The results from the thin section analyses are used to
reconstruct 3-D sandstone models. The reconstruction
algorithm has been described in detail previously40. The
essence of our approach is to build sandstone models which
are analogs of actual sandstones by stochastically model the
results of the main sandstone forming processes sedimentation, compaction, and diagenesis. In the present
work, the modeling of the sedimentation and compaction
processes are identical to that described previously40. The
diagenesis modeling has been extended to include nonuniform quartz cement overgrowth and microporous porelining and pore-filling clays or cements.
In our initial attempt40 at modeling quartz cement
overgrowth, the radii of all the sandgrains were increased
uniformly. As shown by Schwartz and Kimminau41, more
realistic algorithms can be defined by letting the rate of
cement growth depend on the direction of growth. For each
direction r (measured from the grain centre), we define the
length l(r) as the distance between the surface of the original
spherical grain and the surface of its Voronoi polyhedron40,42.
The rate of increase of the distance L(r) between the grain
centre and the pore-grain surface is controlled by a cement
growth exponent α according to the formula41
L(r ) = Ro + min(al α , l )
(1)
where Ro is the radius of the original spherical grain and a
simply controls the amount of cement growth (i.e., the
porosity). The effect of the growth exponent α is illustrated
in Fig. 1. Positive values of α favour growth of cement in the
directions of large l(r) (i.e., the pore bodies). There is a
Fig.1—Cross-sections of computer generated models illustrating the effect of increasing volumes of quartz cement
overgrowth for negative (left) and positive (right) values of the cement growth exponent.
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
clear tendency to form sheet-like pores as the porosity
decreases. This ensures that the pore space remains
interconnected down to very low porosities. Negative values
of α favour growth toward the directions with the smallest
l(r) (i.e., the pore throats). Pore throats thus tend to be
closed off as the porosity decreases. This increases the
tortuosity of the medium and the porosity (i.e., threshold
percolation porosity) at which the connectivity of the pore
space vanishes is higher than that for α > 0. If α = 0, quartz
cement deposits as concentric overgrowth on the grains.
Many varieties of clays or cements may precipitate in the
pore space during the diagenetic stages. Diagenetic clays
clog and sub-divide pore space and generate microporosity.
Pores within the microporosity are typically one or two
orders of magnitude less than the intergranular pores. We
assume that the permeability of these pores is negligible
compared to that of the intergranular pore space.
The morphology of the common diagenetic clays can be
divided into three broad categories: pore-lining, pore-filling,
and pore-bridging. Pore lining clays typically form thin
crystals which grow radially outward from the surface of the
detrital grains. Typical examples are chlorite, illite, and
smectite. Pore-lining clays are modeled by randomly
precipitating clay particles (voxels) on the surfaces of the
detrital grains or the quartz cement. Fig. 2 shows that the
presence of pore-lining clays reduces the size of the
intergranular pores. However, their effect on altering the
effective tortuosity of the medium is small.
Pore-filling clays such as pseudo-hexagonal booklets of
kaolinite are modeled using a clay clustering routine which
causes clay particles (voxels) to precipitate preferentially in
randomly selected pore bodies. Fig. 2 shows that pore-filling
clays tend to form tight clusters of particles which clog or
block off pore throats even at relatively high porosities. The
presence of pore-filling clays thus increases the tortuosity of
the medium significantly.
The morphology of the clay and the total amount of clay
to be precipitated in our models are determined from
petrographical analyses of thin sections. Pore-bridging clays
are currently not implemented.
Pore Network. An example of a reconstructed pore space is
shown in Fig. 3. Although it is possible to perform flow
simulations directly on the chaotic pore space, either by
numerically solving the Navier Stokes equation43,44 or by
applying a Lattice Boltzmann simulation45, it can only be
done at considerable computational expense. It is therefore
convenient to construct a pore network which replicates the
essential features of the pore space that are relevant to fluid
flow. The transformation of the reconstructed pore space into
a pore network and the topological and geometrical
characterisation of the network has been described in detail
previously40.
Pore Shape. A key characteristic of real porous media is
the angular corners of pores. Angular corners retain wetting
fluid and allow two or more fluids to flow simultaneously
Fig.2—Cross-sections of computer generated models illustrating the effect of increasing volumes of pore-lining (left)
and pore-filling (right) clays.
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
For a given G, the minimum and maximum allowed values
of β2 are found from Eq. (4) by setting β2 = β1 and β2 = π/4 β1/2, respectively. In the present work, corner angles are
assigned to pore bodies and throats by first randomly
selecting β2 (β2,min ≤ β2 ≤ β2,max). The remaining angles
may then be determined from Eq. (3).
Flow in the Network
We wish to simulate two-phase flow in our pore networks.
The displacing fluid is injected through an external reservoir
which is connected by pore throats to every pore body on the
inlet side of the network. The displaced fluid escapes
through the outlet face on the opposite side. Periodical
boundary conditions are imposed along the sides parallel to
the main direction of flow.
We assume that capillary forces dominate at the porescale. This is a reasonable assumption for low capillary
number (10-6 or less) processes47 which are typical of most
reservoir displacements. In the following, we present a
precise description of the different pore-scale displacement
mechanisms which may occur in the pore network during
primary drainage and water injection.
Fig. 3—A small cube of a reconstructed pore space. The
linear scale of the model is 100 voxels (0.75mm) on each
side.
through the same pore. Pores which are angular in crosssection are thus a much more realistic model of the pore
structure than the commonly applied cylindrical shape. In the
present work, the shape of every pore body and throat is
described in terms of a dimensionless shape factor G which
is defined as46
G=
A
(2)
s2
Pce =
where A is the average cross-sectional area of the pore body
or throat and s is the corresponding perimeter length. The
area and perimeter length are determined by standard image
analysis techniques. The shape factor replaces the irregular
shape of a pore body or throat by an equivalent irregular
triangular shape. It ranges from zero for a slit shaped pore to
0.048 for an equilateral triangular pore.
It is useful to relate G to the actual shape of the pore body
or throat. From elementary geometry, G may be expressed as
G=
1  3 1
∑
4  i =1 tanβ i
Primary Drainage. Initially, the network is fully saturated
with water and is strongly water wet. Oil then enters the
network representing migration into the reservoir. At every
stage of the process, oil invades the available pore body or
throat with the lowest threshold capillary pressure. This
forms the basis for the invasion percolation algorithm48,49
used previously to model drainage processes.
Threshold or entry capillary pressures are calculated
using the Mayer & Stowe50 and Princen51-53 (MS-P) method.
The details of the calculations are given in Appendix A. The
threshold capillary pressure is governed by both the pore
shape and the receding contact angle θr




−1
(3)
where β is the corner half angle ( β1 ≤ β 2 ≤ β 3 ). Obviously,
a single value of G corresponds to a range of shapes. The
combinations of β1 and β2 which give a constant G is46
sin(2 β1 ) 
sin(2 β1 ) 
 2 +

G=
sin(2 β 2 ) 
2

−2
(4)
γ (1 + 2 πG )cosθ r
Fd (θ r , G )
r
(5)
where r is the inscribed radius of the pore body or throat. In
general, the function Fd is dependent on the particular corner
angles and is thus not universal for a specific G. In practice,
however, Fd varies little for a given G. This is illustrated in
Fig. 4 which shows the full variation in Fd when water is
present in at least one of the corners. For strongly water wet
systems (i.e., θr = 0o), Fd = 1.
Wettability Alteration. When oil initially invades a water
filled pore body or throat, a stable water film protects the
pore surface from wettability change by adsorption. At a
critical capillary pressure, the film collapses to form a
molecular thin film. This allows surface active components
in the oil to adsorb on the pore surface. The capillary
pressure at which the water film ruptures depends on the
curvature of the pore wall and on the shape of the disjoining
pressure isotherm. Kovscek et al.54 present a detailed
analysis of this scenario for star-shaped pores. Blunt32
recently extended the analysis to square pores and presents a
parametric model for the critical capillary pressure at which
the water film collapses. A similar approach is adopted here.
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
contact angle θa is different from θr. This is always true for
mixed wet systems where θa may be much larger than θr.
In this case, there is a range of capillary pressure where
the invading interface remains pinned. As the capillary
pressure drops, the interface remains fixed in place and the
contact angle adjusts to a new value θh. The hinging angle θh
can acquire any value between θr and θa. Water enters the
throat once the capillary pressure is lowered sufficiently that
θa is reached. The calculation of the threshold capillary
pressure for this case is given in Appendix A.
Solutions for the normalised threshold capillary pressure
(PcN = rPce /γ) shows that spontaneous imbibition (PcN > 0)
by piston type displacement always occurs for θa less than
90o and may occur for θa much greater than 90o (Fig. 5). The
maximum advancing angle at which spontaneous imbibition
occurs is given by
G = 0.048
G = 0.03
G = 0.01
G = 0.005
1,5
Fd
1,4
1,3
1,2
1,1
1
0
15
30
45
60
75
90
Receding Angle
Fig. 4—Fd(θr,G) vs. θr when water is present in at least
one of the corners. The results for each G are the
average of 1000 realisations.
For a pore body or throat i, the critical capillary pressure Pc*
is given by
Pc*,i = Π i + Γi xi
(6)
where Π and Γ are input parameters and x varies uniformly
between zero and one. The first term on the right represents
the disjoining pressure whilst the second term represents the
film curvature. If Pc* is exceeded in an oil filled pore body
or throat, the contact angle must change54. We assume that
the new value of θr < 90o - β1. This ensures that water is
always present in at least one of the corners.
Primary drainage and the film rupturing process continues
until a maximum capillary pressure Pc,max is reached. If Pc,max
exceeds Pc* in a pore body or throat, the water film collapses
and the wettability changes. In this case, the corners contain
bulk water and remain water wet. Away from the corners,
adsorption of hydrophobic components make the pore wall
oil wet. We refer to these pore bodies and throats as being
mixed wet. Pore bodies and throats where Pc,max < Pc* remain
water wet.
Water Injection. The capillary pressure drops during water
injection and water first invades the available water wet pore
bodies and throats. Pore-scale displacement mechanisms for
strongly wetting systems have been described by Lenormand
et al.55. There are three types of displacements: piston type,
pore body filling, and snap-off. Below we summarise the
behaviour and the threshold capillary pressures for each of
these displacement in mixed wet pores.
Piston Type. This refers to the displacement of oil from a
pore throat by an invading interface initially located in an
adjoining water filled pore body. If there is no contact angle
hysteresis, the threshold capillary pressure is the same as for
drainage (Eq. (5)). In a more realistic scenario, the advancing
 − 4G ∑ cos(θ r + β i ) 
θ a ,max = cos −1  Nd

 Pc − cosθ r + 12Gsinθ r 
(7)
where PcNd = rPc,max /γ. The maximum advancing angle for
spontaneous imbibition thus depends on Pc,max, G, and θr
which, in turn, determines the residual water saturation in oil
invaded pores. This is illustrated in Fig 6 which shows that
θa,max and therefore spontaneous imbibition decreases when
the residual water saturation is reduced.
If the advancing angle is greater than θa,max, the water
invasion is forced. Forced water invasion is analogous to the
oil drainage process. If θa > 90o + β1, the entry capillary
pressure may be determined from Eq. (5). If θa < 90 o + β1,
the threshold capillary pressure is given by
Pce =
2γ cosθ a
r
(8)
2
Normalise Threshold Pressure
1,6
G = 0.048
G = 0.03
G = 0.01
G = 0.005
1,5
1
0,5
0
0
20
40
60
80
100
120
Advancing Angle
Fig. 5—Normalised threshold pressure for piston
Nd
displacement. The receding angle is zero and Pc = 2.
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
If there is contact angle hysteresis, the AMs remain
pinned at the position established at Pc,max until the hinging
angle in the sharpest corner equals θa. Further decrease in the
capillary pressure causes the AM to advance towards the
centre of the pore space. Eventually, this AM meets another
AM, causing snap-off. If θa < 90o - β1, the curvatures of the
AMs are positive and snap-off occurs at a positive capillary
pressure. The capillary pressure at which this occurs can be
calculated from elementary geometry (Appendix A). In
contrast to piston type invasion, spontaneous imbibition by
snap-off only occurs for θa < 90o (Fig. 7).
Inital Water Saturation
0,6
G = 0.048
G = 0.03
G = 0.01
G = 0.005
0,45
0,3
1
0
90
100
110
120
130
Advancing Angle
Fig. 6—Effect of initial water saturation on the maximum
contact angle at which spontaneous imbibition by piston
type displacement occurs. The receding angle is zero.
Pore body filling. The threshold capillary pressure for
pore body filling is limited by the largest radius of curvature
required to invade the pore body. This radius depends on the
size of the pore body and on the numbers of connecting
throats filled with oil. For a pore body with coordination
number z, there are z-1 such mechanisms, referred to as I1 to
Iz-1. If only one of the connecting throats contain oil (i.e., I1),
the filling of the pore body is similar to that of a piston type
invasion and the threshold capillary pressure is the same as
for the corresponding piston type displacement.
The threshold capillary pressures for the I2 to Iz-1
mechanisms are more complex, especially for the chaotic
networks used in the present work. Blunt32 presents a
parametric model for this and a similar approach is used
here. If θa < θa,max, the mean radius of curvature for filling by
an In mechanism is computed as
Rn =
n
1 

 r p + ∑ bi ri xi 
cosθ a 
i =1

(9)
where rp is the pore body radius, bi are input parameters, ri
are the radii of the oil filled throats, and xi are random
numbers between zero and one. The threshold capillary
pressure is Pc,n= 2γ/Rn. If θa > θa,max, the water invasion is
forced and the threshold capillary pressure is the same as for
the I1 mechanism.
Snap-Off. Snap-off refers to the invasion of an oil filled
pore space by arc menisci (AMs) which always exist in the
corners of oil filled pore bodies and throats. In the absence of
contact angle hysteresis, the AMs advance smoothly along
the pore walls as the capillary pressure drops. At a critical
point, the AMs fuse together and the centre of the pore space
spontaneously fills with water. For a strongly water wet
system, this occurs at a threshold capillary pressure Pce = γ/r.
Normalised Threshold Pressure
0,15
G = 0.048
G = 0.03
G = 0.01
G = 0.005
0,8
0,6
0,4
0,2
0
0
15
30
45
60
75
90
Advancing Angle
Fig. 7— Normalised threshold pressure for snap-off. The
Nd
receding angle is zero and Pc = 2. The results for each
G are the average of 1000 realisations.
If θa > 90o - β1, the curvatures of the AMs are negative
and the invasion is forced. Once the hinging angle in the
sharpest corner has increased to θa, the AM advances
towards the centre of the pore space. This causes the absolute
value of the negative curvature to decrease. The AM is thus
unstable and the pore body or throat immediately fills with
water. The threshold capillary pressure for this process
depends on the curvature of the AM when it begins to move.
It is given by
Pce = Pc ,max
cos(θ a + β1 )
cos(θ r + β 1 )
θa ≤ 180o - β1
(10)
Pce = Pc,max
−1
cos(θ r + β1 )
θa ≥ 180o - β1
(11)
Formation of Oil Films. Piston like advance in a mixed
wet pore body or throat leaves the centre of the pore space
filled with water. If θa > 90o + β1 > θa,max, a film of oil may
be left sandwiched in between the water in the corner(s) and
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
the water in the centre. These films may significantly
increase the connectivity of the oil phase and allow oil
mobility down to very low saturations.
Similar to Blunt32, we assume that the oil film in a corner
is stable until the two oil-water interfaces on either side of
the film meet. The critical capillary pressure at which an oil
film in a corner i collapses is thus given by

Pc
sinβ i
1− d 2

=
− Pc, max cos(θ r + β i )  dcosβ + 1 − d 2 sin 2 β
i
i



 (12)

u=
g=
Fluid Volumes and Conductances
(13)
where rw=γ/Pc is the radius of curvature of the AMs and θ
can be θr, θh, or θa depending on the circumstances. When
oil films are present in the corners of a pore body or throat,
the area occupied by the oil is
3  cos θ cos(θ − β )
π θ − βi

a
a
i
−  a
− 1 − Aw
Ao = rw2 ∑ 
sin β i
2  90
i =1 

(14)
where Aw is given by Eq. (13) and the sum only includes the
corners where oil films exist. The volume of water or oil is
assumed to be the fraction of the cross-section occupied by
the phase multiplied by the total pore body or throat volume.
The volume of trapped fluid is computed using the capillary
pressure at which the fluid was first trapped. The overall
saturation of each phase is found by adding the volume of
each phase in every pore body and throat and dividing by the
total pore volume of the network.
Conductances. The hydraulic conductance of completely
filled pore bodies and throats is computed by assuming a
Poiseuille law relation between the flow rate q and the
pressure gradient ∇P
q = − g∇ P
(15)
At low Reynolds numbers, incompressible flow in a straight
horizontal pore body or throat may be described in
dimensionless form as
∂ 2u ∂ 2u
+
= −1
∂Y 2 ∂ Z 2
3 A 2 G 3r 2 A
=
5µ
20 µ
0,03
0,025
0,02
0,015
0,01
0,005
0
0
where the independent space variables y = Y A and z =
Z A . The dimensionless velocity u is defined as
0,01
0,02
0,03
0,04
0,05
Shape Factor G
2
Fig. 8—Computed dimensionless conductances (gµ/A )
as a function of the shape factor for different triangular
shaped pores.
When water is present as AM in a corner i with a contact
angle less than 90o - βi, the water conductance is
g w, i =
(16)
(18)
If oil occupies the centre of a pore body or throat with water
present as AMs in the corners, the oil conductance is found
from Eq. (18), but with A replaced by the cross-sectional
area occupied by oil.
Dimensionless Conductance
3  cos θ cos(θ + β )
π  θ + β i 
i
− 1 −
Aw = rw2 ∑ 

β
sin i
2
90 
i =1 
(17)
The boundary condition is that u=0 along the walls of the
pore body or throat. Eq. (16) was solved using a standard
finite difference method. Hydraulic conductances were
determined by integrating the computed velocity field over
the pore body or throat cross-section.
Computed conductances for different shaped triangular
pores are shown in Fig. 8. The functional dependency
between g and the shape factor G is almost linear and is
closely approximated by
where d= 2 + cosθa/sinβi.
Pore bodies and throats are assumed to have a triangular
cross-section with area A = r2/4G. When water is present as
AMs in the corners of a pore body or throat, the area
occupied by the water is given by
vx µ
A(− dP / dx )
rw2 Aw,i
C w, i µ
(19)
where Cw is a dimensionless flow resistance factor which
accounts for the reduced water conductivity close to the pore
walls. Numerical solutions of the corner flow problem56
show that Cw depends on the corner geometry, the contact
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
angle, and the boundary condition at the oil-water interface.
The total conductance is the sum of all the corner
conductances.
If the contact angle is greater than 90o - βi, the curvature
of the AM is negative and we do not have an exact
expression for the conductance. In this case, we use an
approximation based on Eq. (19) which may be written as
Aw2 ,i
(20)
C w,i κ i µ
Results
where Cw is evaluated at θ = 0o and κ is given by
κi =
β 
cos β i π 
− 1 − i 
sin β i 2  90 
(21)
If oil films are present in the corners of a pore body or
throat, the oil conductance is found from Eq. (20) but with
Aw replaced by Ao . This is an approximation since the
geometry of the oil film is different to that of a water AM32.
Macroscopic Properties. For laminar flow, the flow rate
of fluid i between two connecting nodes I and J is given by
q i , IJ =
g i , IJ
L IJ
( Pi , I − Pi , J )




(23)
where the subscript t denotes the pore throat. We impose
mass conservation at each pore body, which means that
∑ q i , IJ = 0
Fontainebleau sandstone. The sandstone reconstruction
algorithm was validated by comparing a reconstructed
sample of Fontainebleau sandstone with microtomographic
images of the actual sample. Fig. 9 compares the standard
deviation in local porosity for the two samples. Local
porosity distributions were measured using a moving box
technique with a logarithmic increase in box size. This
ensures that both local and global variations in porosity are
captured. The model closely mimics the actual sample,
except at large box sizes where the variation in porosity is
too small. This suggests that the reconstructed model does
not capture all of the global heterogeneities which are
present in the real sample.
(22)
where LIJ is the spacing between the pore body centres. The
effective conductance gi,IJ is assumed to be the harmonic
mean of the conductances of the throat and the two pore
bodies themselves, i.e.,
LIJ
L
L
1 L
= t +  I + J

g i, IJ
g i ,t 2  g i , I g i , J
(25)
where σw is the electrical conductivity of the fluid which fills
the pore network. The electrical conductivity σ of the pore
network is computed by applying a constant voltage drop
across the network and then use Eq. (22) to find the flow of
current between each pore body. From this one may
calculate the total current and thus the formation factor, FRF
= σw/σ.
(24)
30
25
Standard Deviation (%)
g w, i =
ge = σ w A
Micro-CT
Model
20
15
10
J
5
where J runs over all the pore throats which are connected to
pore body I. Eqs. (22-24) give rise to a set of linear
equations for the pore body pressures.
The permeability of the network is computed by imposing
a constant pressure gradient across the network and let the
system relax using a conjugate gradient method to determine
the pore body pressures. From the pressure distribution we
calculate the total flow rate and thus the permeability using
Darcy’s law. Relative permeabilities are computed similarly.
At various stages of the displacement, we compute the
pressure drop in each phase separately. The saturation and
the total flow rate in each phase are also calculated. Relative
permeabilities may then be determined from Darcy’s law.
Because of the analogy between Poseuille’s law and
Ohm’s law, the flow of electrical current in the network is
also described by Eqs. (22-24), but with pressure replaced by
voltage and hydraulic conductance by electrical conductance.
The electrical conductance ge is assumed to depend only
upon the geometry of the pore body or throat (i.e., pore walls
are insulating) and is given by
0
1
10
100
1000
Box size
Fig. 9—Standard deviation in local porosity for a
reconstructed model and a micro-CT image of a
Fontainebleau sandstone.
The horizontal and vertical two-point correlation
functions were computed as a function of varying box size
and distance, thus covering both local and global
correlations. These correlation functions may be represented
as 2-D surfaces and are shown in Fig. 10. The correlation
functions for the two samples are very similar. Statistical
analysis (ANOVA) shows that there is no significant
differences within the level of noise we experience (1000
replications for each box size-distance combination).
32
16
16
1000
4
0.5
4
8
2
1
2
1
2
8
4
16
32
1
64
1
2
Distance
4
8
16
32
64
Distance
64
64
32
32
16
16
Permeability (mD)
5
0.0
0.05
Box size
8
10000
0.9
5
32
0.9
5
64
0.5
Box size
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
64
Experimental
Model
100
10
0.05
1
1
2
4
8
Distance
16
32
0.0
5
4
0,1
0
2
64
1
0,05
0,1
0,15
0,2
0,25
0,3
Porosity
1
2
4
8
16
32
64
Distance
Fig. 10—Horizontal (top) and vertical (bottom) two-point
correlation functions for the Fontainebleau sandstone.
Micro-CT image to the left, reconstructed model to the
right (1 unit = 7.5 µm).
Bourbie and Zinszer57 performed extensive laboratory
measurements of the hydraulic properties of Fontainebleau
sandstone. The Fontainebleau sandstone is an aeolan fine
grained quartzite. Thin section analyses reveal that it is well
sorted with an average grain size of around 200µm. The
porosity and permeability, however, vary widely. The
variation in these properties is predominantly due to
variation in the volume of quartz cement overgrowth. Using
the reconstructed model as basis, we generated models with
different porosities by varying the amount of quartz cement
overgrowth. The value of the cement growth exponent α was
0.6 for all the models.
The predicted permeability versus porosity trend is
compared with the experimental data of Bourbie and
Zinszer57 in Fig 11. Although the measured permeabilities
span nearly five orders of magnitude, the predicted
permeability versus porosity trend is in good agreement with
the experimental one. Fig 11 shows that there is a change in
the permeability versus porosity trend at approximately 10 to
13% porosity. Our simulation results show that this
corresponds to the porosity at which quartz cement begins to
close off pore throats. This increases the tortuosity of the
medium and the permeability declines more rapidly. Fig 12
compares predicted and measured58,59 formation factors. The
predicted formation factors are also in good agreement with
the measured ones, except for some deviations at high
porosities where we tend to underestimate the formation
factor. These results demonstrate that our sandstone models
and their network representations adequately predict single
phase transport properties for this texturally and diagentically
simple quartz sandstone.
Fig. 11—Comparison between predicted and measured
absolute permeability for Fontainebleau sandstone.
1000
Formation Factor
2
8
0.5
0.5
4
0.9
5
Box size
5
8
0.9
Box size
1
Experimental
Model
100
10
1
0
0,05
0,1
0,15
0,2
0,25
0,3
Porosity
Fig. 12—Comparison between predicted and measured
formation factors for Fontainebleau sandstone.
Bentheimer Sandstone. Thin section images of the
Bentheimer core plugs described earlier were analysed and
used to reconstruct two realisations of each sample. The
main input parameters for the reconstruction algorithm are
summarised in Table 5. Simulated petrophysical properties
for the reconstructed samples are compared with measured
values in Table 6.
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
10
TABLE 5—INPUT PARAMETERS FOR THE SANDSTONE
RECONSTRUCTION ALGORITHM
B2
90-303
2.5
m
0.05
-0.2
0.204
2.55
10.0
B3
90-303
2.5
m
0.05
-0.2
0.201
2.55
10.0
R1
21-74
20.0
p-f
0.0
1.0
0.210
1.02
4.0
R2
21-74
19.0
p-f
0.0
1.0
0.218
1.02
4.0
TABLE 6—COMPARISON BETWEEN PREDICTED AND
MEASURED PETROPHYSICAL PROPERTIES FOR
BENTHEIMER (B) AND RESERVOIR (R) ROCK SAMPLES
Porosity
Sample
B1
B2
B3
R1
R2
Exp.
0.232
0.241
0.237
0.297
0.293
Pred.
0.210
0.219
0.216
0.243
0.245
Permeability (mD)
Exp.
2840
2820
2930
358
326
Pred.
3650
3906
3764
429
472
Form. Factor
Exp.
11.6
12.1
12.0
Pred.
9.4
11.2
11.1
18.6
17.7
The predicted porosities are 2-3 porosity units less than
the plug porosities which were measured with a Helium
porosimeter. The discrepancy between measured and
predicted porosity is most likely due to differences in the
amount of microporosity which is difficult to determine
accurately from thin section analyses. Predicted formation
factors are in good agreement with the experimental data
whilst the predicted permeabilities are approximately 30%
higher than the measured ones. This suggests that we
overestimate the hydraulic conductance of pore bodies and
throats. One possible explanation for this may be that surface
roughness is not accounted for in our expression (Eq. (18))
for the hydraulic conductance.
Primary drainage and waterflood displacements were
simulated on each of the realisations. In all the simulations
the receding contact angle was zero whilst the advancing
contact angle for oil invaded pore bodies and throats was
randomly distributed between 30o and 50o. This is in
agreement with contact angle measurements on quartz60. The
corner resistance factor Cw was calculated assuming a no
flow boundary condition at the oil-water interface56.
Fig. 13 compares simulated and measured drainage
capillary pressures. The calculated capillary pressures are the
average for the six realisations. The predicted capillary
pressure curve is in good agreement with the measured data
and we correctly predict the threshold displacement pressure
and the low connate water saturation. For the high capillary
pressures established at the end of the primary drainage
simulations, oil has invaded almost all of the intergranular
pore space. The connate water saturation is thus mainly made
up of water present in the corners of oil invaded pore bodies
and throats and water present in the microporosity which is
assumed to be impermeable.
Experimental
Predicted
Capillary Pressure (bar)
Parameter
B1
grain radius (µm)
90-303
clay content (wt%) 2.5
clay texture
m
compaction factor
0.05
cement exponent
-0.2
intergranular porosity 0.194
3
size (mm )
2.55
resolution (µm)
10.0
m = mixture
p-f = pore-filling
1
0,1
0,01
0
0,2
0,4
0,6
0,8
1
Sw
Fig. 13—Comparison between predicted and measured
primary drainage capillary pressure for the water wet
Bentheimer sandstone.
Simulated primary drainage and waterflood relative
permeability curves are compared with experimental data in
Fig. 14. Although the predicted results are the average of
only six realisations, the comparison shows that the network
model reproduces the experimental characteristics of the
measured data and correctly predicts the waterflood residual
oil saturation. At low water saturations, the predicted water
permeability is larger than the measured one. This suggest
that we overestimate the water conductance in the corners of
oil invaded pore bodies and throats. The expression for the
corner water conductance (Eq. (19)) assumes a perfectly
sharp corner with smooth pore walls. In reality, the pore
walls are irregular and rough and their curvature is not zero.
Both surface roughness and pore wall curvature can reduce
the water conductance
Reservoir Rock. The main clay component in the
reservoir rock samples is pore-filling kaolinite. In addition,
the samples contain lesser amounts of clay pseudomorphs
after K-feldspar, carbonate cement, pyrite, and trace amounts
of chlorite. For simplicity, we model all of this as pore-filling
clay or cement. Input parameters for the reconstruction
algorithm are given in Table 5. Two realisations of each core
plug were reconstructed. The predicted porosities and
permeabilities are given in Table 6.
The predicted porosities are about five porosity units less
than the Helium porosities. This may be because we
underestimate the amount of microporosity or because the
thin sections are not fully representative of the core plugs
(i.e., heterogeneity). The predicted permeabilities are in fair
agreement with the measured values.
As mentioned before, the reservoir rock samples display a
mixed wettability. Unfortunately, it is not possible to know a
priori what fraction of the pore space which is mixed wet nor
the contact angles which should be assigned to the water wet
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
1
1
krw(experimental)
kro(experimental)
kro(predicted)
krw(predicted)
krw(experimental)
kro(experimental)
kro(predicted)
krw(predicted)
0,8
Relative Permeability
Relative Permeability
0,8
0,6
0,4
0,6
0,4
0,2
0,2
0
0
0
0,2
0,4
0,6
0,8
0
1
0,2
0,4
0,6
0,8
1
Sw
Sw
Fig. 14—Comparison between predicted and measured drainage (left) and waterflood (right) relative permeabilities
for a water wet Bentheimer sandstone.
0,6
0,5
Residual Oil Saturation
and mixed wet regions of the pore space. Instead, we
investigate how the waterflood residual oil saturation Sor
depends on the fraction f of oil invaded pore bodies and
throats which become mixed wet.
Fig. 15 shows how Sor depends on the fraction of mixed
wet pore bodies and throats. In all the simulations, the
advancing contact angle for mixed wet pore bodies and
throats is 180o whilst θa for the water wet regions is 40o. The
receding angle is 20o. For small f, the mixed wet regions of
the pore space do not form a continuos cluster. Oil is thus
trapped in the mixed wet regions of the pore space and Sor
increases almost linearly with f. At a critical f (around 0.62),
mixed wet pore bodies and throats first form a continuous
flow path to the outlet. Forced water injection can then
displace oil and Sor drops sharply. These results are in
qualitative agreement with those reported by Blunt32.
The experimentally determined Sor for the reservoir rock
is 5%. From Fig. 15 this corresponds to f = 0.85. Predicted
waterflood capillary pressure and relative permeability
curves for this case are shown in Fig. 16. The simulated
capillary pressure curve is in fair agreement with the
experimental data except at low oil saturations. At low oil
saturations, the simulated capillary pressure curve displays a
sharper break than the experimental data. This suggests that
the reconstructed sample has a narrower pore size
distribution than the actual sample and that some of the
smaller pores are not included in the model. In order to
capture the presence of these small pores, the resolution of
the reconstructed model must be increased. This, however,
can only be done at considerable computational expense.
0,4
0,3
0,2
0,1
0
0
0,2
0,4
0,6
0,8
1
Fraction Mixed Wet Pores
Fig. 15—Waterflood residual oil saturation for the
reconstructed reservoir rock as a function of the fraction
of pore bodies and throats contacted by oil which
become mixed wet.
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
300
1E+0
Predicted
Exp. (R1)
Exp. (R2)
1E-1
Relative Permeability
Capillary Pressure (kPa)
200
100
0
-100
krw (sim)
kro (sim)
kro (R1)
kro (R2)
1E-2
1E-3
1E-4
-200
1E-5
-300
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
Sw
Sw
Fig. 16—Comparison between simulated and measured waterflood capillary pressures and relative permeabilities for
a mixed wet reservoir rock.
The agreement between the predicted and measured oil
relative permeabilities is encouraging. In spite of the
necessarily approximate nature for the distribution of water
wet and mixed wet pores as well as the uncertain estimates
for both the contact angles and the oil film conductance, the
predicted and measured oil relative permeability curves
display the same generic behaviour. Both the experiments
and the simulations show that continuos oil films allow low
oil saturations to be reached during forced water injection.
However, the oil relative permeability is very low.
The fair agreement between the predicted and measured
transport properties for the three different reconstructed
sandstones is encouraging. Although we need to examine a
much larger group of petrographically and petrophysically
heterogeneous rock samples, these preliminary results
cautiously suggest that network modeling techniques may be
used to a priori predict single and multiphase transport
properties for real rocks.
Conclusions
1. Petrographical data obtained from image analysis of
two-dimensional thin sections are used to reconstruct model
sandstones which give a realistic description of the complex
microstructure exhibited in real sandstones.
2. Image analysis techniques are used to transform the
reconstructed pore space into a pore network which can be
used as input to a network simulator.
3. Predicted permeabilities and formation factors of a
reconstructed Fontainebleau sandstone model correspond
well with published data over a wide range of porosity.
4. Predicted primary drainage and waterflood relative
permeability curves for a water wet Bentheimer sandstone
are in good agreement with experimental data.
5. Simulated waterflood capillary pressures and oil
relative permeabilities for a mixed wet reservoir rock are in
fair agreement with experimental measurements.
Nomenclature
A = cross-sectional area
a = parameter controlling the amount of cement
growth
b = input parameter, Eq. (9)
C = corner resistance factor
FRF = formation factor
f = fraction of mixed wet pore bodies and throats
G = pore body or throat shape factor
g = conductance
k = permeability
kr = relative permeability
L = length
Lb = distance between pinned contact line and corner
l = distance between surface of grain and its
Voronoi polyhedron
P = pressure
Pc = capillary pressure
Pc* = threshold capillary pressure for water film
collapse, Eq. (6)
Pc,max = maximum capillary pressure reached during
primary drainage
q = volumetric flowrate
Rn = radius of curvature for pore body filling
r = radius
Sor = residual oil saturation
s = perimeter length
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
u=
v=
α=
β=
Γ=
φ=
γ=
µ=
Π=
θ=
ρ=
σ=
dimensionless velocity
velocity
cement growth exponent
corner half angle
input parameter, Eq. (6)
porosity
interfacial tension
viscosity
parameter controling water film collapse
contact angle
density
electrical conductivity
Subscripts
a = advancing
d = drainage
e = electrical
eff = effective
h = hinging
min = minimum
max = maximum
o = oil
p = pore body
pd = primary drainage
pt = piston type
s = solid
r = receding
t = pore throat
w = water
Superscripts
e = entry
d = drainage
N = normalised
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Acknowledgments
The authors would like to acknowledge Den norske stats
oljeselskap a.s. (Statoil) for granting permission to publish
this paper. We are indebted to Svein H. Midtlyng for
performing the experimental work. We also graciously thank
David Stern, Exxon Research Production Company, for
providing us with the microtomography data.
19.
20.
21.
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Appendix A - Threshold Pressures
Mason and Morrow46 applied the MS-P method to derive a
general expression in terms of G for the drainage threshold
capillary pressure in strongly wetted triangular pores. This
analysis was later extended to include the effect of contact
angle and contact angle hysteresis in equilateral triangular
pores61. In the present work, we generalise this model to any
EXTENDING PREDICTIVE CAPABILITIES TO NETWORK MODELS
shape triangular pores. Our computations are similar to those
reported by Ma et al.62 for the displacement curvature of a
meniscus in regular polygonal tubes.
Drainage. Capillary forces prevent oil from spontaneously
entering water filled throats. Oil can only enter an available
throat if the capillary pressure exceeds the threshold
capillary pressure. At the threshold capillary pressure, oil
enters the throat with a fixed curvature and displaces water
from the central part of the throat, leaving some of the water
as arc menisci (AMs) in the corners. In the absence of
gravity, the curvature of these AMs is the same as the
curvature of the invading interface.
The MS-P method for calculating threshold capillary
pressures relies on equating the curvature, 1/rd, of the AMs
to the curvature of the invading interface. If the AMs are
displaced a small distance dx, the work of the displacement
must be balanced by the change in surface free energy
Pc Aeff dx = ( Lowγ ow + Los γ os − Los γ ws )dx
(A-1)
where the subscript s denotes the solid, Aeff is the effective
area occupied by oil, Los is the length of the solid wall in
contact with oil, and Low is the perimeter length of the AMs.
From Young’s equation, γos - γws = γowcosθr and Eq. (A-1)
simplifies to
Leff
Pc
L + Los cos θ r
1
=
= ow
=
γ ow rd
Aeff
Aeff
(A-2)
The curvature of the invading interface at the condition for
invasion is thus given as the ratio of the effective perimeter
to the effective area. Aeff, Low, and Los are readily determined
from elementary geometry and are given by
3  cos θ cos(θ + β )
π  θ + β i 
r
r
i
− 1 − r
Aeff = A − rd2 ∑ 

sin β i
2
90 
i =1 
=
r2
− rd2 S1
4G
(A - 3)
Los =
3 cos(θ + β )
r
r
r
i
− 2rd ∑
=
− 2rd S 2
2G
sin β i
2G
i =1
(A-4)
Low =
2πrd 3
∑ (90 − θ r − β i ) = rd S 3
180 i =1
(A-5)
rd =
4GD
(A-7)
where the function Fd(θr,G) is given by
Fd (θ r , G ) =
1 + 1 + 4GD / cos 2 θ r
1 + 2 πG
(A-8)
In general, Fd(θr,G) is dependent on the particular corner
angles and is not universal for a specific G. However, if
AMs are present in all the corners, the expression for D
becomes
cos 2 θ r
 θ 
D = π 1 − r  + 3 sin θ r cos θ r −
4G
 60 
(A-9)
D is thus only dependent on θr and Fd(θr,G) is universal for a
particular G.
Piston Type. If there is contact angle hysteresis, the
threshold capillary pressure for piston type invasion during
water injection is different from that during drainage. The
invading interface enters the throat once the curvature is
lowered sufficiently that θa is reached. The oil-water
interfaces of the AMs in the throat remain pinned at the
position Lb established at the end of the primary drainage,
i.e.,
Lbi = r pd
cos(θ r + β i )
sin β i
(A-10)
where rpd = γ/Pc,max and Lbi is the distance between the
pinned contact line and the corner i. As the capillary pressure
drops, the hinging angle of the pinned AMs adjusts to give
the same curvature as for the invading interface. Provided
that θa is not too large, the invading interface meets these
AMs at zero contact angle. The radius of curvature rpt of the
AMs may be calculated by equating rpt to Aeff/Leff. The
effective area Aeff is given by Eq. (A-3), but with rpt
substituted for rd and θr replaced by the hinging angle θh,i
 r pd

θ h,i = cos −1 
cos(θ r + β i )  − β i
 r pt



(A-11)
The effective perimeter Peff is given by
After a bit of algebra, we obtain a quadratic expression for rd
which has the solution
r cos θ r (−1 ± 1 + 4GD / cos 2 θ r )
r
r
= Pc
= cos θ r (1 + 2 πG ) Fd (θ r , G )
γ ow
rd
3
 r2
 2πr pt 3
− 2 ∑ Lbi  +
Leff = cos θ a 
∑α i
i =1
 2G
 180 i =1
(A-12)
where the angle αi is given by
(A-6)
where D = S1 - 2S2cosθr + S3. The radius of curvature rd is
given by the valid root (smaller radius than the inscribed
radius r) of Eq. (A-6) and may be expressed as
 L sin β i
α i = sin −1  bi

r pt





(A-13)
P.E. ØREN, S. BAKKE, AND O.J. ARNTZEN
The above equations constitute a non-linear system of
equations which can be solved numerically for the radius of
curvature rpt and thus the threshold capillary pressure Pc =
γ/rpt. The maximum advancing angle at which spontaneous
imbibition occurs is given by Eq. (7) and corresponds to the
limit when Leff = 0. If θa > θa,max, the threshold capillary
pressure may be determined from Eqs. (5) or (8).
Snap-Off. If θa < 90o - β1, snap-off occurs when the
advancing AM in the sharpest corner meets one of the other
AMs. For small values of θa, all the AMs advance towards
the centre of the pore space at θa and snap-off occurs when
the AMs in the two sharpest corners meet. This occurs at a
threshold capillary pressure
Pc =
γ
r

2 sin θ a
 cos θ a −
cot β1 + cot β 2




(A-14)
For larger values of θa, the AM in the largest corner may
remain pinned whilst the two other AMs advance at θa.
Snap-off then occurs either when the two advancing AMs
meet or when the AM in the sharpest corner meets the
pinned AM. The threshold capillary pressure for the first
case is given by Eq. (A-14) whilst the threshold capillary
pressure in the second case is
Pc =
γ
r
 cos θ a cot β1 − sin θ a + cos θ h3 cot β 3 − sin θ h3 




cot β1 + cot β 3


(A-15)
The event which actually takes place is the one occurring at
the highest capillary pressure. Because the hinging angle
depends on the prevailing capillary pressure, the above
equation can not be solved analytically. If θa > 90o - β1, the
curvatures of the AMs are negative and the threshold
capillary pressure is given by Eq. (10) or (11).
SI Metric Conversion Factors
cp x 1.0*
E–03 = Pa. s
md x 9.869223 E–04 = µm 2
*Conversion factor is exact
SPEJ
Pål-Eric Øren, SPE, is a research scientist at Statoil
Research Centre, Postuttak, N-7005 Trondheim, Norway,
email: [email protected]. His research interests include
physics of multiphase flow in porous media and reservoir
engineering. Øren holds a BS in chemical engineering from
the University of Michigan, and a PhD in petroleum
engineering from the University of New South Wales. Stig
Bakke is a geologist at Statoil Research Centre, Postuttak,
N-7005 Trondheim, Norway, email: [email protected]. His
research interests include micro-scale reservoir description,
numerical modeling of geological process, and 2-D and 3-D
image analysis. Before joining Statoil, he worked at IKU,
Trondheim. Bakke holds a MS in geology from NTH. Ole
Jakob Arntzen is a research scientist at Statoil Research
Centre, Postuttak, N-7005 Trondheim, Norway, email:
[email protected]. His research interests include multiphase
flow in porous media, probability and statistics. Arntzen
holds a MS in applied mathematics. Before joining Statoil he
worked at the Norwegian Defense Research Establishment.